Search Results (24)

This paper studies constrained Newtonian fluid flows through porous media, accounting for the drag effect on the fluid, modeled using a Mixture Theory perspective and a constitutive relation for the pressure—namely, a continuous and differentiable function of the saturation that ensures always preserving the problem hyperbolicity. The pressure equation also permits an ultra-small porous matrix supersaturation (that is controlled) and the transition from unsaturated to saturated flow (and vice versa). The mathematical model gives rise to a nonlinear, non-homogeneous hyperbolic system. Its numerical simulation combines Glimm’s method with an operator-splitting strategy to account for the Darcy and Forchheimer terms that cause the system’s non-homogeneity. Despite the Glimm method’s proven convergence, it is not adequate to approximate non-homogeneous hyperbolic systems unless combined with an operator-splitting technique. Although other approaches have already addressed this problem, the novelty is combining Glimm’s method with operator-splitting to account for linear and nonlinear drag effects. Glimm’s scheme marches in time using a formerly selected number of associated Riemann problems. The constitutive relation for the pressure—an increasing function of the saturation, with the first derivative also increasing, convex, and positive, enables us to obtain explicit expressions for the Riemann invariants. The results show the influence of the Darcy and Forchheimer drag terms on the flow. Full article

The necessity of understanding and simulating hydrological phenomena as well as their interactions and the effect of anthropogenic and climate conditions on the ecosystem have encouraged researchers for years to investigate the moisture transfer in soil. Considering the moisture transfer as an isothermal phenomenon might cause a wrong estimation due to the non-isothermal nature of the moisture movement in porous media. Hygrothermal (coupled heat and moisture transfer) models are quite diverse and are the engine of the various hygrothermal software tools used to analyze the heat and moisture in building envelopes, drying technologies, and many other applications. This paper is a literature survey conducted to provide an overview on the classical hygrothermal models to address the historical perspectives on these models. First, it investigated, from a historical point of view, the challenges behind the development of hygrothermal models as unsaturated flow theories, beginning with Buckingham theory. The non-isothermal nature of moisture was the starting point for researchers to deal with new challenges during mathematical modeling and experimental analysis. In general, the theory of coupled heat and moisture transfer first developed by J.R. Philip and De Vries and the authors in the mid-1950s inspired the novel hygrothermal models, including Sophocleous and Milly’s model, Rode’s model, Künzel’s model, and Grunewal’s model. In a parallel of hygrothermal model developments, the models of Whitaker and Luikov can also be classified as hygrothermal models; they were mostly applied in modeling the phenomenon of drying. The study highlights the application of hygrothermal models in building physics and gathered a summary of international efforts such as Annex 24, Annex 41, and the HAMSTAD project and advancements performed from the classical dew point or steady-state Glaser method. Moreover, this study emphasizes the advantages of the standard of EN 15026 and limitations of the Glaser method. To sum up, hygrothermal models are still under development based on various assumptions of moisture driving potentials and transfer coefficients. Full article

Contrary to the continuum hypothesis, which averages water flow across the entire domain, including both grains and pores, the line-element model concentrates unsaturated flow in the pore space in the intermediate region of horizontal and vertical channels. The flux equivalent principle is used to deduce the equivalent unsaturated hydraulic conductivity, the flow velocity and the continuity equations. It is found that the relative hydraulic conductivities derived from the line-element model and the continuum model are identical. The continuity equations in the two models are also similar, except that the coefficient in the water content term is half that in the line-element model. Thus, the unsaturated flow problem in porous media is transformed into a one-dimensional problem. A dimension-reduced finite line-element method is proposed that includes a complementary algorithm for Signorini’s-type boundary conditions involving the seepage-face boundary and the infiltration boundary. The validity of the proposed model is then proved by good agreement with analytical, experimental and simulated results for one-dimensional infiltration in a vertical soil column, unsaturated flow in a sand flume with drainage tunnels, and transient unsaturated flow water-table recharge in a soil slab, respectively. In general, the proposed method has good computational efficiency, especially for smaller mesh sizes and short time intervals. Full article

The sustainable exploitation of groundwater resources is a multifaceted and complex problem, which is controlled, among many other factors and processes, by water flow in porous soils and sediments. Modeling water flow in unsaturated, non-deformable porous media is commonly based on a partial differential equation, which translates the mass conservation principle into mathematical terms. Such an equation assumes that the variation of the volumetric water content ($\theta$) in the medium is balanced by the net flux of water flow, i.e., the divergence of specific discharge, if source/sink terms are negligible. Specific discharge is in turn related to the matric potential (h), through the non-linear Darcy–Buckingham law. The resulting equation can be rewritten in different ways, in order to express it as a partial differential equation where a single physical quantity is considered to be a dependent variable. Namely, the most common instances are the Fokker–Planck Equation (for $\theta$), and the Richards Equation (for h). The other two forms can be given for generalized matric flux potential ($\Phi$) and for hydraulic conductivity (K). The latter two cases are shown to limit the non-linearity to multiplicative terms for an exponential K-to-h relationship. Different types of boundary conditions are examined for the four different formalisms. Moreover, remarks given on the physico-mathematical properties of the relationships between K, h, and $\theta$ could be useful for further theoretical and practical studies. Full article

This work uses a mixture theory approach to describe kinematically constrained flows through porous media using an adequate constitutive relation for pressure that preserves the problem hyperbolicity even when the flow becomes saturated. This feature allows using the same mathematical tool for handling unsaturated and saturated flows. The mechanical model can represent the saturated–unsaturated transition and vice-versa. The constitutive relation for pressure is a continuous and differentiable function of saturation: an increasing function with a strictly convex, increasing, and positive first derivative. This significant characteristic permits the fluid to establish a tiny controlled supersaturation of the porous matrix. The associated Riemann problem’s complete solution is addressed in detail, with explicit expressions for the Riemann invariants. Glimm’s semi-analytical scheme advances from a given instant to a subsequent one, employing the associated Riemann problem solution for each two consecutive time steps. The simulations employ a variation in Glimm’s scheme, which uses the mean of four independent sequences for each considered time, ensuring computational solutions with reliable positions of rarefaction and shock waves. The results permit verifying this significant characteristic. Full article

In order to reduce the numerical difficulty of the 3D transient free surface flow problems in porous media, a line element method is proposed by dimension reduction. Different from the classical continuum-based methods, homogeneous permeable pores in the control volume are conceptualized by a 3D orthogonal network of tubes. To obtain the same hydraulic solution with the continuum model, the equivalent formulas of flow velocity, continuity equation and transient free surface boundary are derivable from the principle of flow balance. In the solution space of transient free surface flow, the 3D problem is transformed into 1D condition, and then a finite element algorithm is simply deduced. The greatest advantage of the line element method is line integration instead of volume/surface integration, which has dramatically decreased the integration difficulty across the jump free surface. Through the analysis of transient free surface flow in the unconfined aquifer, trapezoidal dam, sand flume and wells, the transient free surface locations predicted from the proposed line element method generally agree well with the analytical, experimental and other numerical data in the available literatures, the numerical efficiency can also be well guaranteed. Furthermore, the hydraulic anisotropy has significant effect on the evolution of free surface locations and the shape of depression cones in spatial. The line element method can be expanded to model the 3D unsaturated seepage flow, two-phase flow and thermos problems in porous media because of the similarity between the similarity of Darcy’s law, Buckingham Law and Fourier’s law. Full article

The computational cost of approximating the Richards equation for water flow in unsaturated porous media is a major challenge, especially for tasks that require repetitive simulations. Data-driven modeling offers a faster and more efficient way to estimate soil moisture dynamics, significantly reducing computational costs. Typically, data-driven models use one-dimensional vectors to represent soil moisture at specific points or as a time series. However, an alternative approach is to use images that capture the distribution of porous media characteristics as input, allowing for the estimation of the two-dimensional soil moisture distribution using a single model. This approach, known as image-to-image regression, provides a more explicit consideration of heterogeneity in the porous domain but faces challenges due to increased input–output dimensionality. Deep neural networks (DNNs) provide a solution to tackle the challenge of high dimensionality. Particularly, encoder–decoder convolutional neural networks (ED-CNNs) are highly suitable for addressing this problem. In this study, we aim to assess the precision of ED-CNNs in predicting soil moisture distribution based on porous media characteristics and also investigate their effectiveness as an optimizer for inverse modeling. The study introduces several novelties, including the application of ED-CNNs to forward and inverse modeling of water flow in unsaturated porous media, performance evaluation using numerical model-generated and laboratory experimental data, and the incorporation of image stacking to account for transient moisture distribution. A drainage experiment conducted on a sandbox flow tank filled with monodisperse quartz sand was employed as the test case. Monte Carlo simulation with a numerical model was employed to generate data for training and validation of the ED-CNN. Additionally, the ED-CNN optimizer was validated using images obtained through non-intrusive photographic imaging. The results show that the developed ED-CNN model provides accurate approximations, addressing the high-dimensionality problem of image-to-image regression. The data-driven model predicted soil moisture with an R² score of over 91%, while the ED-CNN optimizer achieved an R² score of over 89%. The study highlights the potential of ED-CNNs as reliable and efficient tools for both forward and inverse modeling in the analysis of unsaturated flow. Full article

Understanding the displacement of the resident wetting fluid in porous media is crucial to the remediation strategy. When pollutants or nutrients are dissolved in the surface wetting fluid and enter the unsaturated zone, the resident wetting fluid in the porous system may remain or be easily flushed out and finally arrive in the groundwater. The fate and transport of the resident wetting fluid determine the policy priorities on soil or groundwater. In this study, the displacement of the resident wetting fluid by the invading wetting fluid in porous media was simulated using direct numerical simulation (DNS). Based on the simulations of the displacements in porous media, the effect of the non-wetting fluid on the displacement was evaluated by observation and quantification, which were difficult to achieve in laboratory experiments. The result can also explain the unknown phenomenon in previous column experiments, namely that the old water is continuously released from the unsaturated porous media even after a long period of flushing with the new water. The effects of the interfacial tension, contact angle, and injection rate, which affected the immiscible fluid–fluid flow pattern, were also evaluated. Since pollutants dissolved in the wetting fluid could change the physical properties of the wetting fluid, the interfacial tensions of the resident wetting fluid and the invading wetting fluid were set separately in the simulation. Moreover, our simulation demonstrated that the consecutive drainage–imbibition cycles could improve the displacement of the resident wetting fluid in porous media. The successful simulation in this study implied that this method can be applied to predict other immiscible fluid–fluid flow in natural or industrial processes. Full article

In this paper, we introduce peridynamic theory and its application to Richards’ equation with a piecewise smooth initial condition. Peridynamic theory is a non-local continuum theory that models the deformation and failure of materials. Richards’ equation describes the unsaturated flow of water through porous media, and it plays an essential role in many applications, such as groundwater management, soil science, and environmental engineering. We develop a peridynamic formulation of Richards’ equation that includes the effect of peridynamic forces and a piecewise smooth initial condition, further introducing a non-standard symmetric influence function to describe such peridynamic interactions, which turns out to provide beneficial effects from a numerical point of view. Moreover, we implement a numerical scheme based on Chebyshev polynomials and symmetric Gauss–Lobatto nodes, providing a powerful spectral method able to capture singularities and critical issues of Richards’ equation with piecewise smooth initial conditions. We also present numerical simulations that illustrate the performance of the proposed approach. In particular, we perform a computational investigation into the spatial order of convergence, showing that, despite the discontinuity in the initial condition, the order of convergence is retained. Full article

The seismoelectric effect of porous media is the main basis for seismoelectric logging. At present, most of the studies on the seismoelectric effect in unsaturated porous media adopt the model of pores with continuous distribution of gas and liquid. There is a lack of theoretical research on the micro mechanism of the seismoelectric effect of unsaturated porous media with discrete gas phase, and the existing studies do not consider the effect of the electric double layer at the gas–liquid interface on the seismoelectric effect. Based on the capillary model, this work adopted the gas phase discrete model, combined the electric double layer theory and the seepage principle, considered the effect of electric double layer at the pore wall and the gas–liquid interface, and studied the micro principle of the seismoelectric effect of unsaturated porous media. Firstly, we studied the variation of gas–water two-phase flow pattern with saturation in unsaturated pores, then proposed the equivalent principle of series circuits, deduced the effective streaming current and conductance of a pore containing multiple bubbles, and then deduced the streaming potential coupling coefficient in the unsaturated pores. We also studied the effect of pore parameters such as saturation, pore size, bubble spacing, pore fluid viscosity, and salinity on the streaming potential coupling coefficient. The results show that the streaming potential coupling coefficient first increases and then decreases with the decrease in saturation, which is the same as the trend measured in Allègre’s experiment, and provide a theoretical explanation for the non-monotonic change in the coupling coefficient with saturation in unsaturated porous media. Full article

Gravitational fingering often occurs for water flow in unsaturated porous media. This paper reviews a recent effort in developing a macroscopic theory to describe the gravitational fingering flow of water in homogeneous and unsaturated soils, based on the optimality principle that water flows in unsaturated soils in such a manner that the generated flow patterns correspond to the minimum global flow resistance. The key difference between the new theory and the conventional unsaturated flow theories is that the hydraulic conductivity in the new theory is not only related to water saturation or capillary pressure, but also proportional to a power function of water flux, because the water flux is closely related to the fingering flow patterns and the power function allows for large hydraulic conductivities at locations where water fluxes are large as well to minimize the global flow resistance. The resultant relationship for the fraction of fingering flow zone is compared with that obtained from a parallel effort based on the fractal nature of fingering flow patterns. The relationships from the two efforts are found to be essentially identical for gravity-dominated water flow in unsaturated soils and can both be expressed as a power function of the water saturation. This work also demonstrates that the theoretical values for the exponent of the power function vary in a relatively narrow range between 0.75 and 0.80 for most soils, which is supported by observations from previous field tests. This remarkable finding makes it easy to apply the new theory to field sites where experimental data are not readily available for estimating the exponent value. The potential limitations of the theory and the suggested future research topics in the area are also discussed. Full article

The water retention curve, which relates the matric potential, ψ, to the water content, θ, is essential to describe the flow processes in the unsaturated zone and provides useful information for environmental and engineering applications. There are few studies devoted to measuring the rock water retention curves due to the rock’s tightness, which makes it more technically difficult to use specific methods. In this study, we tested four different methods to measure water retention curves of two lithotypes of carbonate porous rocks with the aim to find the most effective to be applied to rock samples. Suction table, evaporation, Quasi-Steady Centrifuge, and WP4-T dewpoint potentiameter methods have been applied. The Quasi-Steady Centrifuge method proved to be the only one capable of determining water retention curves in the entire water content range and capturing the bimodality of the tested media with respect to the other methods. The measured water retention data were fitted with HYPROP-FIT software that allows us to accurately describe the WRCs and obtain critical parameters for the numerical simulation of flow and transport through the vadose zone, which plays a key role in various environmental issues. Full article

Determining the constitutive properties that describe the incipient hydraulic behavior of the materials, including the matrix domains and the distribution of macro and micropores, is crucial to analyzing the preferential water flow in saturated soils, k_s, and unsaturated, k_u. This study focused on determining the hydraulic conductivity in porous media under total and partial saturation conditions. The infiltration characteristics of three reconstituted soils were evaluated using five suction ranges employing conventional permeameters, an automated dual system, and mini-disk infiltrometers. The experimental cycles were carried out in granular soils with mixtures of diatomaceous soils, iron oxide (Fe₂O₃), and calcium carbonate (CaCO₃) in 5–40% proportions. The differences between the granular microstructures of each material and the different hydraulic interaction mechanisms (suctione levels) significantly affected the values of k_s and k_u and the coupling between the pore domains and the defined water regime. Additionally, a lower impact was observed in the data set exposed to higher percentages of Fe₂O₃ and CaCO₃ in different suction ranges, mainly due to a tension effect (meniscus) generated by suction in the granular skeleton. Since both parameters are mutually correlated and have a similar impact between methods and soil cores, k_s and k_u must be optimized simultaneously in each mechanism analyzed. The main findings of this work result in the confirmation that the unsaturated permeability decreases as suction is imposed on the sample. As well as the addition of different materials with Particle Size Distribution finer than the base sample, it also reveals a reduction in hydraulic conductivity, both saturated and unsaturated. Full article

Surface-active solutes that exist in the subsurface either naturally (humic acid) or as a result of anthropogenic activities (alcohols, surfactants, PFAS) alter the hydraulic and geotechnical properties of the unsaturated porous media. The alteration of properties is the result of concentration-dependent surface tension, and/or density, and the contact angle effects. These effects are manifested in the form of changes in water retention and conduction and changes in the suction component of the shear strength. Differences in the spatial distribution of these solutes in the subsurface result in capillary pressure gradients causing flow perturbations. Conceptual and numerical models to understand the effects of these solutes require concentration-dependent consideration of surface tension, density, and the contact angle effects on hydraulic and geotechnical properties of porous media. Capillary rise experiments have been carried out to either quantify the effect of surface-active solutes on the height of capillary rise or to determine the concentration-dependent contact angle changes due to salinity of the pore water. This paper provides a comprehensive review of the literature on capillary rise experiments and how they can potentially be used to characterize the hydraulic and geotechnical properties of unsaturated porous media affected by surface-active solutes. Full article

Water flow in porous media is one of many phenomena in nature that can demonstrate both simple and complex behaviors. A soil–water retention curve (SWRC) is needed to characterize this flow properly. This curve relates the soil water content and the matric potential (or porepressure), being fundamental for simulating unsaturated soil behaviors. This article proposes a new model based on simple assumptions regarding the saturated and unsaturated branches of soil–water retention curves. Despite its simplicity, the modeling capability of the proposed SWRC is shown for two types of soil. This new SWRC is obtained as a logistic function after solving an ordinary differential equation (ODE). This ODE can also be solved numerically using the Finite Difference Method (FDM), which indicates that the discrete version of the SWRC can be represented as the logistic map for specific parameters. On the other hand, this discrete representation is known to encompass chaotic and fractal behaviors. This link is used to investigate the stability and convergence of the FDM scheme, indicating that for values pre-bifurcation, both the FDM and the analytical solution of the ODE represent the new SWRC. This way, the present paper is the first step to better understating how a chaotic framework could be related to SWRCs and geotechnics in general. Full article